Question

what is the half - life, in seconds, of a 100g sample of n - 16 that decays to 12.5g of n - 16 in 21.6 seconds?

Explanation:

Step1: Calculate the number of half - lives

The formula for radioactive decay is \(N = N_0\times(\frac{1}{2})^n\), where \(N\) is the final amount, \(N_0\) is the initial amount, and \(n\) is the number of half - lives.
Given \(N_0=100g\), \(N = 12.5g\), we substitute into the formula:
\(12.5=100\times(\frac{1}{2})^n\)
\(\frac{12.5}{100}=(\frac{1}{2})^n\)
\(0.125 = (\frac{1}{2})^n\)
Since \(\frac{1}{8}=(\frac{1}{2})^3\), so \(n = 3\)

Step2: Calculate the half - life

We know that the total time \(t = 21.6s\) and the number of half - lives \(n = 3\)
The formula for half - life \(T=\frac{t}{n}\)
Substitute \(t = 21.6s\) and \(n = 3\) into the formula:
\(T=\frac{21.6}{3}\)
\(T = 7.2s\)

Answer:

\(7.2s\)